VandenBroeck and Keller, Equation (20)

Percentage Accurate: 7.1% → 99.1%

Time: 19.2s

Alternatives: 6

Speedup: 4.7×

• could not determine a ground truth

  1. f = 1478615189.1152885

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{4} \cdot f\\ t_1 := e^{t\_0}\\ t_2 := e^{-t\_0}\\ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{t\_1 + t\_2}{t\_1 - t\_2}\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (/ PI 4.0) f)) (t_1 (exp t_0)) (t_2 (exp (- t_0))))
   (- (* (/ 1.0 (/ PI 4.0)) (log (/ (+ t_1 t_2) (- t_1 t_2)))))))

C

double code(double f) {
	double t_0 = (((double) M_PI) / 4.0) * f;
	double t_1 = exp(t_0);
	double t_2 = exp(-t_0);
	return -((1.0 / (((double) M_PI) / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
}

Java

public static double code(double f) {
	double t_0 = (Math.PI / 4.0) * f;
	double t_1 = Math.exp(t_0);
	double t_2 = Math.exp(-t_0);
	return -((1.0 / (Math.PI / 4.0)) * Math.log(((t_1 + t_2) / (t_1 - t_2))));
}

Python

def code(f):
	t_0 = (math.pi / 4.0) * f
	t_1 = math.exp(t_0)
	t_2 = math.exp(-t_0)
	return -((1.0 / (math.pi / 4.0)) * math.log(((t_1 + t_2) / (t_1 - t_2))))

Julia

function code(f)
	t_0 = Float64(Float64(pi / 4.0) * f)
	t_1 = exp(t_0)
	t_2 = exp(Float64(-t_0))
	return Float64(-Float64(Float64(1.0 / Float64(pi / 4.0)) * log(Float64(Float64(t_1 + t_2) / Float64(t_1 - t_2)))))
end

MATLAB

function tmp = code(f)
	t_0 = (pi / 4.0) * f;
	t_1 = exp(t_0);
	t_2 = exp(-t_0);
	tmp = -((1.0 / (pi / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]}, Block[{t$95$1 = N[Exp[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-t$95$0)], $MachinePrecision]}, (-N[(N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * N[Log[N[(N[(t$95$1 + t$95$2), $MachinePrecision] / N[(t$95$1 - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]]

Initial Program: 7.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{4} \cdot f\\ t_1 := e^{t\_0}\\ t_2 := e^{-t\_0}\\ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{t\_1 + t\_2}{t\_1 - t\_2}\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (/ PI 4.0) f)) (t_1 (exp t_0)) (t_2 (exp (- t_0))))
   (- (* (/ 1.0 (/ PI 4.0)) (log (/ (+ t_1 t_2) (- t_1 t_2)))))))

C

double code(double f) {
	double t_0 = (((double) M_PI) / 4.0) * f;
	double t_1 = exp(t_0);
	double t_2 = exp(-t_0);
	return -((1.0 / (((double) M_PI) / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
}

Java

public static double code(double f) {
	double t_0 = (Math.PI / 4.0) * f;
	double t_1 = Math.exp(t_0);
	double t_2 = Math.exp(-t_0);
	return -((1.0 / (Math.PI / 4.0)) * Math.log(((t_1 + t_2) / (t_1 - t_2))));
}

Python

def code(f):
	t_0 = (math.pi / 4.0) * f
	t_1 = math.exp(t_0)
	t_2 = math.exp(-t_0)
	return -((1.0 / (math.pi / 4.0)) * math.log(((t_1 + t_2) / (t_1 - t_2))))

Julia

function code(f)
	t_0 = Float64(Float64(pi / 4.0) * f)
	t_1 = exp(t_0)
	t_2 = exp(Float64(-t_0))
	return Float64(-Float64(Float64(1.0 / Float64(pi / 4.0)) * log(Float64(Float64(t_1 + t_2) / Float64(t_1 - t_2)))))
end

MATLAB

function tmp = code(f)
	t_0 = (pi / 4.0) * f;
	t_1 = exp(t_0);
	t_2 = exp(-t_0);
	tmp = -((1.0 / (pi / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]}, Block[{t$95$1 = N[Exp[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-t$95$0)], $MachinePrecision]}, (-N[(N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * N[Log[N[(N[(t$95$1 + t$95$2), $MachinePrecision] / N[(t$95$1 - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]]

Alternative 1: 99.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\log \tanh \left(\pi \cdot \left(f \cdot 0.25\right)\right)}{\pi}}{0.25} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (/ (/ (log (tanh (* PI (* f 0.25)))) PI) 0.25))

C

double code(double f) {
	return (log(tanh((((double) M_PI) * (f * 0.25)))) / ((double) M_PI)) / 0.25;
}

Java

public static double code(double f) {
	return (Math.log(Math.tanh((Math.PI * (f * 0.25)))) / Math.PI) / 0.25;
}

Python

def code(f):
	return (math.log(math.tanh((math.pi * (f * 0.25)))) / math.pi) / 0.25

Julia

function code(f)
	return Float64(Float64(log(tanh(Float64(pi * Float64(f * 0.25)))) / pi) / 0.25)
end

MATLAB

function tmp = code(f)
	tmp = (log(tanh((pi * (f * 0.25)))) / pi) / 0.25;
end

Wolfram

code[f_] := N[(N[(N[Log[N[Tanh[N[(Pi * N[(f * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] / 0.25), $MachinePrecision]

Derivation

1. Initial program 8.9%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l/ 8.9%

      \[\leadsto -\color{blue}{\frac{1 \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)}{\frac{\pi}{4}}} \]

4. Applied egg-rr 99.4%

   \[\leadsto -\color{blue}{\frac{-\log \tanh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}{\pi \cdot 0.25}} \]
5. Step-by-step derivation

   1. distribute-frac-neg 99.4%

      \[\leadsto -\color{blue}{\left(-\frac{\log \tanh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}{\pi \cdot 0.25}\right)} \]

   2. distribute-neg-frac2 99.4%

      \[\leadsto -\color{blue}{\frac{\log \tanh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}{-\pi \cdot 0.25}} \]

   3. *-commutative 99.4%

      \[\leadsto -\frac{\log \tanh \color{blue}{\left(f \cdot \left(\pi \cdot 0.25\right)\right)}}{-\pi \cdot 0.25} \]

   4. distribute-rgt-neg-in 99.4%

      \[\leadsto -\frac{\log \tanh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{\color{blue}{\pi \cdot \left(-0.25\right)}} \]

   5. metadata-eval 99.4%

      \[\leadsto -\frac{\log \tanh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{\pi \cdot \color{blue}{-0.25}} \]

   6. *-commutative 99.4%

      \[\leadsto -\frac{\log \tanh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{\color{blue}{-0.25 \cdot \pi}} \]

6. Simplified 99.4%

   \[\leadsto -\color{blue}{\frac{\log \tanh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{-0.25 \cdot \pi}} \]
7. Step-by-step derivation

   1. neg-mul-1 99.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{\log \tanh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{-0.25 \cdot \pi}} \]

   2. associate-*r/ 99.4%

      \[\leadsto \color{blue}{\frac{-1 \cdot \log \tanh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{-0.25 \cdot \pi}} \]

   3. times-frac 99.4%

      \[\leadsto \color{blue}{\frac{-1}{-0.25} \cdot \frac{\log \tanh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{\pi}} \]

   4. metadata-eval 99.4%

      \[\leadsto \color{blue}{4} \cdot \frac{\log \tanh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{\pi} \]

   5. metadata-eval 99.4%

      \[\leadsto \color{blue}{\frac{1}{0.25}} \cdot \frac{\log \tanh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{\pi} \]

   6. times-frac 99.4%

      \[\leadsto \color{blue}{\frac{1 \cdot \log \tanh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{0.25 \cdot \pi}} \]

   7. *-un-lft-identity 99.4%

      \[\leadsto \frac{\color{blue}{\log \tanh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}}{0.25 \cdot \pi} \]

   8. *-commutative 99.4%

      \[\leadsto \frac{\log \tanh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{\color{blue}{\pi \cdot 0.25}} \]

   9. associate-/r* 99.4%

      \[\leadsto \color{blue}{\frac{\frac{\log \tanh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{\pi}}{0.25}} \]

   10. associate-*r* 99.4%

       \[\leadsto \frac{\frac{\log \tanh \color{blue}{\left(\left(f \cdot \pi\right) \cdot 0.25\right)}}{\pi}}{0.25} \]

   11. *-commutative 99.4%

       \[\leadsto \frac{\frac{\log \tanh \left(\color{blue}{\left(\pi \cdot f\right)} \cdot 0.25\right)}{\pi}}{0.25} \]

   12. associate-*l* 99.4%

       \[\leadsto \frac{\frac{\log \tanh \color{blue}{\left(\pi \cdot \left(f \cdot 0.25\right)\right)}}{\pi}}{0.25} \]

8. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\frac{\frac{\log \tanh \left(\pi \cdot \left(f \cdot 0.25\right)\right)}{\pi}}{0.25}} \]
9. Add Preprocessing

Alternative 2: 97.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 1.26:\\ \;\;\;\;\left(\log \left(\frac{4}{\pi}\right) - \log f\right) \cdot \frac{-4}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\pi \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 1.26)
   (* (- (log (/ 4.0 PI)) (log f)) (/ -4.0 PI))
   (/ 0.0 (* PI -0.25))))

C

double code(double f) {
	double tmp;
	if (f <= 1.26) {
		tmp = (log((4.0 / ((double) M_PI))) - log(f)) * (-4.0 / ((double) M_PI));
	} else {
		tmp = 0.0 / (((double) M_PI) * -0.25);
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (f <= 1.26) {
		tmp = (Math.log((4.0 / Math.PI)) - Math.log(f)) * (-4.0 / Math.PI);
	} else {
		tmp = 0.0 / (Math.PI * -0.25);
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if f <= 1.26:
		tmp = (math.log((4.0 / math.pi)) - math.log(f)) * (-4.0 / math.pi)
	else:
		tmp = 0.0 / (math.pi * -0.25)
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (f <= 1.26)
		tmp = Float64(Float64(log(Float64(4.0 / pi)) - log(f)) * Float64(-4.0 / pi));
	else
		tmp = Float64(0.0 / Float64(pi * -0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	tmp = 0.0;
	if (f <= 1.26)
		tmp = (log((4.0 / pi)) - log(f)) * (-4.0 / pi);
	else
		tmp = 0.0 / (pi * -0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := If[LessEqual[f, 1.26], N[(N[(N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision] - N[Log[f], $MachinePrecision]), $MachinePrecision] * N[(-4.0 / Pi), $MachinePrecision]), $MachinePrecision], N[(0.0 / N[(Pi * -0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if f < 1.26000000000000001

   1. Initial program 8.4%

      \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{1}{\mathsf{expm1}\left(\left(0.5 \cdot \pi\right) \cdot f\right)}\right) \cdot \frac{-4}{\pi}} \]
   4. Add Preprocessing

   5. Taylor expanded in f around 0 97.9%

      \[\leadsto \color{blue}{\left(\log \left(\frac{4}{\pi}\right) + -1 \cdot \log f\right)} \cdot \frac{-4}{\pi} \]
   6. Step-by-step derivation

      1. mul-1-neg 97.9%

         \[\leadsto \left(\log \left(\frac{4}{\pi}\right) + \color{blue}{\left(-\log f\right)}\right) \cdot \frac{-4}{\pi} \]

      2. unsub-neg 97.9%

         \[\leadsto \color{blue}{\left(\log \left(\frac{4}{\pi}\right) - \log f\right)} \cdot \frac{-4}{\pi} \]

   7. Simplified 97.9%

      \[\leadsto \color{blue}{\left(\log \left(\frac{4}{\pi}\right) - \log f\right)} \cdot \frac{-4}{\pi} \]

   if 1.26000000000000001 < f

   1. Initial program 24.1%

      \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l/ 24.1%

         \[\leadsto -\color{blue}{\frac{1 \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)}{\frac{\pi}{4}}} \]

   4. Applied egg-rr 96.3%

      \[\leadsto -\color{blue}{\frac{-\log \tanh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}{\pi \cdot 0.25}} \]
   5. Step-by-step derivation

      1. distribute-frac-neg 96.3%

         \[\leadsto -\color{blue}{\left(-\frac{\log \tanh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}{\pi \cdot 0.25}\right)} \]

      2. distribute-neg-frac2 96.3%

         \[\leadsto -\color{blue}{\frac{\log \tanh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}{-\pi \cdot 0.25}} \]

      3. *-commutative 96.3%

         \[\leadsto -\frac{\log \tanh \color{blue}{\left(f \cdot \left(\pi \cdot 0.25\right)\right)}}{-\pi \cdot 0.25} \]

      4. distribute-rgt-neg-in 96.3%

         \[\leadsto -\frac{\log \tanh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{\color{blue}{\pi \cdot \left(-0.25\right)}} \]

      5. metadata-eval 96.3%

         \[\leadsto -\frac{\log \tanh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{\pi \cdot \color{blue}{-0.25}} \]

      6. *-commutative 96.3%

         \[\leadsto -\frac{\log \tanh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{\color{blue}{-0.25 \cdot \pi}} \]

   6. Simplified 96.3%

      \[\leadsto -\color{blue}{\frac{\log \tanh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{-0.25 \cdot \pi}} \]

   8. Applied egg-rr 86.2%

      \[\leadsto -\frac{\log \color{blue}{\left(\frac{\sqrt{\tanh \left(\pi \cdot \left(f \cdot 0.25\right)\right)}}{\sqrt{\tanh \left(\pi \cdot \left(f \cdot 0.25\right)\right)}}\right)}}{-0.25 \cdot \pi} \]
   9. Step-by-step derivation

      1. *-inverses 86.2%

         \[\leadsto -\frac{\log \color{blue}{1}}{-0.25 \cdot \pi} \]

   10. Simplified 86.2%

       \[\leadsto -\frac{\log \color{blue}{1}}{-0.25 \cdot \pi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 1.26:\\ \;\;\;\;\left(\log \left(\frac{4}{\pi}\right) - \log f\right) \cdot \frac{-4}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\pi \cdot -0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 1.26:\\ \;\;\;\;\frac{-4 \cdot \log \left(\frac{4}{\pi \cdot f}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\pi \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 1.26) (/ (* -4.0 (log (/ 4.0 (* PI f)))) PI) (/ 0.0 (* PI -0.25))))

C

double code(double f) {
	double tmp;
	if (f <= 1.26) {
		tmp = (-4.0 * log((4.0 / (((double) M_PI) * f)))) / ((double) M_PI);
	} else {
		tmp = 0.0 / (((double) M_PI) * -0.25);
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (f <= 1.26) {
		tmp = (-4.0 * Math.log((4.0 / (Math.PI * f)))) / Math.PI;
	} else {
		tmp = 0.0 / (Math.PI * -0.25);
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if f <= 1.26:
		tmp = (-4.0 * math.log((4.0 / (math.pi * f)))) / math.pi
	else:
		tmp = 0.0 / (math.pi * -0.25)
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (f <= 1.26)
		tmp = Float64(Float64(-4.0 * log(Float64(4.0 / Float64(pi * f)))) / pi);
	else
		tmp = Float64(0.0 / Float64(pi * -0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	tmp = 0.0;
	if (f <= 1.26)
		tmp = (-4.0 * log((4.0 / (pi * f)))) / pi;
	else
		tmp = 0.0 / (pi * -0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := If[LessEqual[f, 1.26], N[(N[(-4.0 * N[Log[N[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision], N[(0.0 / N[(Pi * -0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if f < 1.26000000000000001

   1. Initial program 8.4%

      \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{1}{\mathsf{expm1}\left(\left(0.5 \cdot \pi\right) \cdot f\right)}\right) \cdot \frac{-4}{\pi}} \]
   4. Add Preprocessing

   5. Taylor expanded in f around 0 97.8%

      \[\leadsto \log \color{blue}{\left(\frac{4}{f \cdot \pi}\right)} \cdot \frac{-4}{\pi} \]
   6. Step-by-step derivation

      1. associate-*r/ 97.9%

         \[\leadsto \color{blue}{\frac{\log \left(\frac{4}{f \cdot \pi}\right) \cdot -4}{\pi}} \]

   7. Applied egg-rr 97.9%

      \[\leadsto \color{blue}{\frac{\log \left(\frac{4}{f \cdot \pi}\right) \cdot -4}{\pi}} \]

   if 1.26000000000000001 < f

   1. Initial program 24.1%

      \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l/ 24.1%

         \[\leadsto -\color{blue}{\frac{1 \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)}{\frac{\pi}{4}}} \]

   4. Applied egg-rr 96.3%

      \[\leadsto -\color{blue}{\frac{-\log \tanh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}{\pi \cdot 0.25}} \]
   5. Step-by-step derivation

      1. distribute-frac-neg 96.3%

         \[\leadsto -\color{blue}{\left(-\frac{\log \tanh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}{\pi \cdot 0.25}\right)} \]

      2. distribute-neg-frac2 96.3%

         \[\leadsto -\color{blue}{\frac{\log \tanh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}{-\pi \cdot 0.25}} \]

      3. *-commutative 96.3%

         \[\leadsto -\frac{\log \tanh \color{blue}{\left(f \cdot \left(\pi \cdot 0.25\right)\right)}}{-\pi \cdot 0.25} \]

      4. distribute-rgt-neg-in 96.3%

         \[\leadsto -\frac{\log \tanh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{\color{blue}{\pi \cdot \left(-0.25\right)}} \]

      5. metadata-eval 96.3%

         \[\leadsto -\frac{\log \tanh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{\pi \cdot \color{blue}{-0.25}} \]

      6. *-commutative 96.3%

         \[\leadsto -\frac{\log \tanh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{\color{blue}{-0.25 \cdot \pi}} \]

   6. Simplified 96.3%

      \[\leadsto -\color{blue}{\frac{\log \tanh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{-0.25 \cdot \pi}} \]

   8. Applied egg-rr 86.2%

      \[\leadsto -\frac{\log \color{blue}{\left(\frac{\sqrt{\tanh \left(\pi \cdot \left(f \cdot 0.25\right)\right)}}{\sqrt{\tanh \left(\pi \cdot \left(f \cdot 0.25\right)\right)}}\right)}}{-0.25 \cdot \pi} \]
   9. Step-by-step derivation

      1. *-inverses 86.2%

         \[\leadsto -\frac{\log \color{blue}{1}}{-0.25 \cdot \pi} \]

   10. Simplified 86.2%

       \[\leadsto -\frac{\log \color{blue}{1}}{-0.25 \cdot \pi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 1.26:\\ \;\;\;\;\frac{-4 \cdot \log \left(\frac{4}{\pi \cdot f}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\pi \cdot -0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.7% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 1.26:\\ \;\;\;\;\frac{-4}{\pi} \cdot \log \left(\frac{4}{\pi \cdot f}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\pi \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 1.26) (* (/ -4.0 PI) (log (/ 4.0 (* PI f)))) (/ 0.0 (* PI -0.25))))

C

double code(double f) {
	double tmp;
	if (f <= 1.26) {
		tmp = (-4.0 / ((double) M_PI)) * log((4.0 / (((double) M_PI) * f)));
	} else {
		tmp = 0.0 / (((double) M_PI) * -0.25);
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (f <= 1.26) {
		tmp = (-4.0 / Math.PI) * Math.log((4.0 / (Math.PI * f)));
	} else {
		tmp = 0.0 / (Math.PI * -0.25);
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if f <= 1.26:
		tmp = (-4.0 / math.pi) * math.log((4.0 / (math.pi * f)))
	else:
		tmp = 0.0 / (math.pi * -0.25)
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (f <= 1.26)
		tmp = Float64(Float64(-4.0 / pi) * log(Float64(4.0 / Float64(pi * f))));
	else
		tmp = Float64(0.0 / Float64(pi * -0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	tmp = 0.0;
	if (f <= 1.26)
		tmp = (-4.0 / pi) * log((4.0 / (pi * f)));
	else
		tmp = 0.0 / (pi * -0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := If[LessEqual[f, 1.26], N[(N[(-4.0 / Pi), $MachinePrecision] * N[Log[N[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.0 / N[(Pi * -0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if f < 1.26000000000000001

   1. Initial program 8.4%

      \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{1}{\mathsf{expm1}\left(\left(0.5 \cdot \pi\right) \cdot f\right)}\right) \cdot \frac{-4}{\pi}} \]
   4. Add Preprocessing

   5. Taylor expanded in f around 0 97.8%

      \[\leadsto \log \color{blue}{\left(\frac{4}{f \cdot \pi}\right)} \cdot \frac{-4}{\pi} \]

   if 1.26000000000000001 < f

   1. Initial program 24.1%

      \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l/ 24.1%

         \[\leadsto -\color{blue}{\frac{1 \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)}{\frac{\pi}{4}}} \]

   4. Applied egg-rr 96.3%

      \[\leadsto -\color{blue}{\frac{-\log \tanh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}{\pi \cdot 0.25}} \]
   5. Step-by-step derivation

      1. distribute-frac-neg 96.3%

         \[\leadsto -\color{blue}{\left(-\frac{\log \tanh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}{\pi \cdot 0.25}\right)} \]

      2. distribute-neg-frac2 96.3%

         \[\leadsto -\color{blue}{\frac{\log \tanh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}{-\pi \cdot 0.25}} \]

      3. *-commutative 96.3%

         \[\leadsto -\frac{\log \tanh \color{blue}{\left(f \cdot \left(\pi \cdot 0.25\right)\right)}}{-\pi \cdot 0.25} \]

      4. distribute-rgt-neg-in 96.3%

         \[\leadsto -\frac{\log \tanh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{\color{blue}{\pi \cdot \left(-0.25\right)}} \]

      5. metadata-eval 96.3%

         \[\leadsto -\frac{\log \tanh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{\pi \cdot \color{blue}{-0.25}} \]

      6. *-commutative 96.3%

         \[\leadsto -\frac{\log \tanh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{\color{blue}{-0.25 \cdot \pi}} \]

   6. Simplified 96.3%

      \[\leadsto -\color{blue}{\frac{\log \tanh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{-0.25 \cdot \pi}} \]

   8. Applied egg-rr 86.2%

      \[\leadsto -\frac{\log \color{blue}{\left(\frac{\sqrt{\tanh \left(\pi \cdot \left(f \cdot 0.25\right)\right)}}{\sqrt{\tanh \left(\pi \cdot \left(f \cdot 0.25\right)\right)}}\right)}}{-0.25 \cdot \pi} \]
   9. Step-by-step derivation

      1. *-inverses 86.2%

         \[\leadsto -\frac{\log \color{blue}{1}}{-0.25 \cdot \pi} \]

   10. Simplified 86.2%

       \[\leadsto -\frac{\log \color{blue}{1}}{-0.25 \cdot \pi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 1.26:\\ \;\;\;\;\frac{-4}{\pi} \cdot \log \left(\frac{4}{\pi \cdot f}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\pi \cdot -0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 3.1% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \log 0 \end{array} \]

FPCore

(FPCore (f) :precision binary64 (log 0.0))

C

double code(double f) {
	return log(0.0);
}

Fortran

real(8) function code(f)
    real(8), intent (in) :: f
    code = log(0.0d0)
end function

Java

public static double code(double f) {
	return Math.log(0.0);
}

Python

def code(f):
	return math.log(0.0)

Julia

function code(f)
	return log(0.0)
end

MATLAB

function tmp = code(f)
	tmp = log(0.0);
end

Wolfram

code[f_] := N[Log[0.0], $MachinePrecision]

Derivation

1. Initial program 8.9%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l/ 8.9%

      \[\leadsto -\color{blue}{\frac{1 \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)}{\frac{\pi}{4}}} \]

4. Applied egg-rr 99.4%

   \[\leadsto -\color{blue}{\frac{-\log \tanh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}{\pi \cdot 0.25}} \]
5. Step-by-step derivation

   1. distribute-frac-neg 99.4%

      \[\leadsto -\color{blue}{\left(-\frac{\log \tanh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}{\pi \cdot 0.25}\right)} \]

   2. distribute-neg-frac2 99.4%

      \[\leadsto -\color{blue}{\frac{\log \tanh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}{-\pi \cdot 0.25}} \]

   3. *-commutative 99.4%

      \[\leadsto -\frac{\log \tanh \color{blue}{\left(f \cdot \left(\pi \cdot 0.25\right)\right)}}{-\pi \cdot 0.25} \]

   4. distribute-rgt-neg-in 99.4%

      \[\leadsto -\frac{\log \tanh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{\color{blue}{\pi \cdot \left(-0.25\right)}} \]

   5. metadata-eval 99.4%

      \[\leadsto -\frac{\log \tanh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{\pi \cdot \color{blue}{-0.25}} \]

   6. *-commutative 99.4%

      \[\leadsto -\frac{\log \tanh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{\color{blue}{-0.25 \cdot \pi}} \]

6. Simplified 99.4%

   \[\leadsto -\color{blue}{\frac{\log \tanh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{-0.25 \cdot \pi}} \]
7. Step-by-step derivation

   1. *-commutative 99.4%

      \[\leadsto -\frac{\log \tanh \color{blue}{\left(\left(\pi \cdot 0.25\right) \cdot f\right)}}{-0.25 \cdot \pi} \]

   2. tanh-def-a 8.9%

      \[\leadsto -\frac{\log \color{blue}{\left(\frac{e^{\left(\pi \cdot 0.25\right) \cdot f} - e^{-\left(\pi \cdot 0.25\right) \cdot f}}{e^{\left(\pi \cdot 0.25\right) \cdot f} + e^{-\left(\pi \cdot 0.25\right) \cdot f}}\right)}}{-0.25 \cdot \pi} \]

   3. div-sub 8.8%

      \[\leadsto -\frac{\log \color{blue}{\left(\frac{e^{\left(\pi \cdot 0.25\right) \cdot f}}{e^{\left(\pi \cdot 0.25\right) \cdot f} + e^{-\left(\pi \cdot 0.25\right) \cdot f}} - \frac{e^{-\left(\pi \cdot 0.25\right) \cdot f}}{e^{\left(\pi \cdot 0.25\right) \cdot f} + e^{-\left(\pi \cdot 0.25\right) \cdot f}}\right)}}{-0.25 \cdot \pi} \]

   4. associate-*l* 8.8%

      \[\leadsto -\frac{\log \left(\frac{e^{\color{blue}{\pi \cdot \left(0.25 \cdot f\right)}}}{e^{\left(\pi \cdot 0.25\right) \cdot f} + e^{-\left(\pi \cdot 0.25\right) \cdot f}} - \frac{e^{-\left(\pi \cdot 0.25\right) \cdot f}}{e^{\left(\pi \cdot 0.25\right) \cdot f} + e^{-\left(\pi \cdot 0.25\right) \cdot f}}\right)}{-0.25 \cdot \pi} \]

   5. exp-prod 8.4%

      \[\leadsto -\frac{\log \left(\frac{\color{blue}{{\left(e^{\pi}\right)}^{\left(0.25 \cdot f\right)}}}{e^{\left(\pi \cdot 0.25\right) \cdot f} + e^{-\left(\pi \cdot 0.25\right) \cdot f}} - \frac{e^{-\left(\pi \cdot 0.25\right) \cdot f}}{e^{\left(\pi \cdot 0.25\right) \cdot f} + e^{-\left(\pi \cdot 0.25\right) \cdot f}}\right)}{-0.25 \cdot \pi} \]

   6. *-commutative 8.4%

      \[\leadsto -\frac{\log \left(\frac{{\left(e^{\pi}\right)}^{\color{blue}{\left(f \cdot 0.25\right)}}}{e^{\left(\pi \cdot 0.25\right) \cdot f} + e^{-\left(\pi \cdot 0.25\right) \cdot f}} - \frac{e^{-\left(\pi \cdot 0.25\right) \cdot f}}{e^{\left(\pi \cdot 0.25\right) \cdot f} + e^{-\left(\pi \cdot 0.25\right) \cdot f}}\right)}{-0.25 \cdot \pi} \]

   7. cosh-undef 8.4%

      \[\leadsto -\frac{\log \left(\frac{{\left(e^{\pi}\right)}^{\left(f \cdot 0.25\right)}}{\color{blue}{2 \cdot \cosh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}} - \frac{e^{-\left(\pi \cdot 0.25\right) \cdot f}}{e^{\left(\pi \cdot 0.25\right) \cdot f} + e^{-\left(\pi \cdot 0.25\right) \cdot f}}\right)}{-0.25 \cdot \pi} \]

   8. associate-*l* 8.4%

      \[\leadsto -\frac{\log \left(\frac{{\left(e^{\pi}\right)}^{\left(f \cdot 0.25\right)}}{2 \cdot \cosh \color{blue}{\left(\pi \cdot \left(0.25 \cdot f\right)\right)}} - \frac{e^{-\left(\pi \cdot 0.25\right) \cdot f}}{e^{\left(\pi \cdot 0.25\right) \cdot f} + e^{-\left(\pi \cdot 0.25\right) \cdot f}}\right)}{-0.25 \cdot \pi} \]

   9. *-commutative 8.4%

      \[\leadsto -\frac{\log \left(\frac{{\left(e^{\pi}\right)}^{\left(f \cdot 0.25\right)}}{2 \cdot \cosh \left(\pi \cdot \color{blue}{\left(f \cdot 0.25\right)}\right)} - \frac{e^{-\left(\pi \cdot 0.25\right) \cdot f}}{e^{\left(\pi \cdot 0.25\right) \cdot f} + e^{-\left(\pi \cdot 0.25\right) \cdot f}}\right)}{-0.25 \cdot \pi} \]

8. Applied egg-rr 3.1%

   \[\leadsto -\frac{\log \color{blue}{\left(\frac{{\left(e^{\pi}\right)}^{\left(f \cdot 0.25\right)}}{2 \cdot \cosh \left(\pi \cdot \left(f \cdot 0.25\right)\right)} - \frac{{\left(e^{\pi}\right)}^{\left(f \cdot 0.25\right)}}{2 \cdot \cosh \left(\pi \cdot \left(f \cdot 0.25\right)\right)}\right)}}{-0.25 \cdot \pi} \]
9. Step-by-step derivation

   1. +-inverses 3.1%

      \[\leadsto -\frac{\log \color{blue}{0}}{-0.25 \cdot \pi} \]

10. Simplified 3.1%

    \[\leadsto -\frac{\log \color{blue}{0}}{-0.25 \cdot \pi} \]
11. Step-by-step derivation

    1. add-sqr-sqrt 0.0%

       \[\leadsto \color{blue}{\sqrt{-\frac{\log 0}{-0.25 \cdot \pi}} \cdot \sqrt{-\frac{\log 0}{-0.25 \cdot \pi}}} \]

    2. sqrt-unprod 0.7%

       \[\leadsto \color{blue}{\sqrt{\left(-\frac{\log 0}{-0.25 \cdot \pi}\right) \cdot \left(-\frac{\log 0}{-0.25 \cdot \pi}\right)}} \]

    3. sqr-neg 0.7%

       \[\leadsto \sqrt{\color{blue}{\frac{\log 0}{-0.25 \cdot \pi} \cdot \frac{\log 0}{-0.25 \cdot \pi}}} \]

    4. sqrt-unprod 0.7%

       \[\leadsto \color{blue}{\sqrt{\frac{\log 0}{-0.25 \cdot \pi}} \cdot \sqrt{\frac{\log 0}{-0.25 \cdot \pi}}} \]

    5. add-log-exp 0.7%

       \[\leadsto \color{blue}{\log \left(e^{\sqrt{\frac{\log 0}{-0.25 \cdot \pi}} \cdot \sqrt{\frac{\log 0}{-0.25 \cdot \pi}}}\right)} \]

    6. add-sqr-sqrt 0.7%

       \[\leadsto \log \left(e^{\color{blue}{\frac{\log 0}{-0.25 \cdot \pi}}}\right) \]

    7. div-inv 0.7%

       \[\leadsto \log \left(e^{\color{blue}{\log 0 \cdot \frac{1}{-0.25 \cdot \pi}}}\right) \]

    8. exp-to-pow 0.7%

       \[\leadsto \log \color{blue}{\left({0}^{\left(\frac{1}{-0.25 \cdot \pi}\right)}\right)} \]

    9. associate-/r* 0.7%

       \[\leadsto \log \left({0}^{\color{blue}{\left(\frac{\frac{1}{-0.25}}{\pi}\right)}}\right) \]

    10. metadata-eval 0.7%

        \[\leadsto \log \left({0}^{\left(\frac{\color{blue}{-4}}{\pi}\right)}\right) \]

12. Applied egg-rr 0.7%

    \[\leadsto \color{blue}{\log \left({0}^{\left(\frac{-4}{\pi}\right)}\right)} \]
13. Step-by-step derivation

    1. pow-base-0 3.1%

       \[\leadsto \log \color{blue}{0} \]

14. Simplified 3.1%

    \[\leadsto \color{blue}{\log 0} \]
15. Add Preprocessing

Alternative 6: 4.9% accurate, 106.4× speedup

Math

\[\begin{array}{l} \\ \frac{0}{\pi \cdot -0.25} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (/ 0.0 (* PI -0.25)))

C

double code(double f) {
	return 0.0 / (((double) M_PI) * -0.25);
}

Java

public static double code(double f) {
	return 0.0 / (Math.PI * -0.25);
}

Python

def code(f):
	return 0.0 / (math.pi * -0.25)

Julia

function code(f)
	return Float64(0.0 / Float64(pi * -0.25))
end

MATLAB

function tmp = code(f)
	tmp = 0.0 / (pi * -0.25);
end

Wolfram

code[f_] := N[(0.0 / N[(Pi * -0.25), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.9%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l/ 8.9%

      \[\leadsto -\color{blue}{\frac{1 \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)}{\frac{\pi}{4}}} \]

4. Applied egg-rr 99.4%

   \[\leadsto -\color{blue}{\frac{-\log \tanh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}{\pi \cdot 0.25}} \]
5. Step-by-step derivation

   1. distribute-frac-neg 99.4%

      \[\leadsto -\color{blue}{\left(-\frac{\log \tanh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}{\pi \cdot 0.25}\right)} \]

   2. distribute-neg-frac2 99.4%

      \[\leadsto -\color{blue}{\frac{\log \tanh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}{-\pi \cdot 0.25}} \]

   3. *-commutative 99.4%

      \[\leadsto -\frac{\log \tanh \color{blue}{\left(f \cdot \left(\pi \cdot 0.25\right)\right)}}{-\pi \cdot 0.25} \]

   4. distribute-rgt-neg-in 99.4%

      \[\leadsto -\frac{\log \tanh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{\color{blue}{\pi \cdot \left(-0.25\right)}} \]

   5. metadata-eval 99.4%

      \[\leadsto -\frac{\log \tanh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{\pi \cdot \color{blue}{-0.25}} \]

   6. *-commutative 99.4%

      \[\leadsto -\frac{\log \tanh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{\color{blue}{-0.25 \cdot \pi}} \]

6. Simplified 99.4%

   \[\leadsto -\color{blue}{\frac{\log \tanh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{-0.25 \cdot \pi}} \]

8. Applied egg-rr 5.4%

   \[\leadsto -\frac{\log \color{blue}{\left(\frac{\sqrt{\tanh \left(\pi \cdot \left(f \cdot 0.25\right)\right)}}{\sqrt{\tanh \left(\pi \cdot \left(f \cdot 0.25\right)\right)}}\right)}}{-0.25 \cdot \pi} \]
9. Step-by-step derivation

   1. *-inverses 5.4%

      \[\leadsto -\frac{\log \color{blue}{1}}{-0.25 \cdot \pi} \]

10. Simplified 5.4%

    \[\leadsto -\frac{\log \color{blue}{1}}{-0.25 \cdot \pi} \]

11. Final simplification 5.4%

    \[\leadsto \frac{0}{\pi \cdot -0.25} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024180 
(FPCore (f)
  :name "VandenBroeck and Keller, Equation (20)"
  :precision binary64
  (- (* (/ 1.0 (/ PI 4.0)) (log (/ (+ (exp (* (/ PI 4.0) f)) (exp (- (* (/ PI 4.0) f)))) (- (exp (* (/ PI 4.0) f)) (exp (- (* (/ PI 4.0) f)))))))))